A Unified Framework on Global Stability and Lyapunov Dimension of Lur'e Systems

The equivalence between local and global characteristics of Lur'e systems is investigated. Historically, such problems date back to Vyshnegradskii's conjecture on Watt governors and Eden's conjecture on Lorenz attractors. In the present paper, we develop a unified framework for stability and dimension analyses. This is motivated by the recent works on hidden oscillations and their relations with absolute stability theory. We combine an energy perspective in Lyapunov analysis and a linearization approach in contraction analysis to study global stability and Lyapunov dimension. Notably, we allow a gap between the storage function and the Lyapunov function by utilizing what we call a differential Lyapunov function of the Leonov form. Our framework is also less conservative in the sense that the exact global stability condition and the exact Lyapunov dimension can be characterized. The effectiveness of our method is demonstrated through the Lorenz attractor.